Most of the proposed and experimentally studied 3D HOT crystalline materials turn out to be of the translationally-active type. For example, elemental Bi exhibits a double band inversion at the \(C_3\) -symmetric \(T\) -point in the BZ and supports mixed electronic topology manifesting through coexisting gapless hinge and Dirac surface modes [1]}, [2]}, [3]}. Our general mechanism thus implies that an edge dislocation with the Burgers vector in the \((111)\)  direction parallel to the HOT mass domain wall (see, Fig. 1c in Ref. [1]}), so that also \({\Phi _{\rm dis}}=\pi \) , features gapless modes, protected by \(C_3\)  and time-reversal symmetries. A screw dislocation oriented in the same direction should host one-dimensional gapless (gapped) states if the \((111)\)  surface is gapless (gapped). Similarly, for a recently proposed HOT insulator in Zr(TiH\(_2\) )\(_2\) , with band-inversion away from the \(\Gamma \)  point and the gapless modes along all the edges in the cubic geometry [5]}, we predict gapless (gapped) modes in the core of an edge (a screw) dislocation with the Burgers vector along a principal crystal axis. Finally, the candidate HOT insulators Bi\(_4\) X\(_4\) , with X=Br,I, [5]}, [7]}, [8]}, [9]} feature band inversions at \(R\)  and \(M=(\pi /a,\pi /a,0)\)  points in the BZ, and hence the dislocations should host the one-dimensional modes, following the above general rule.
